Jacobi's theorem (geometry)

Adjacent colored angles are equal in measure. The point N is the Jacobi point for triangle ABC and these angles.

In plane geometry, a Jacobi point is a point in the Euclidean plane determined by a triangle ABC and a triple of angles α, β, and γ. This information is sufficient to determine three points X, Y, and Z such that ∠ZAB = ∠YAC = α, ∠XBC = ∠ZBA = β, and ∠YCA = ∠XCB = γ. Then, by a theorem of Karl Friedrich Andreas Jacobi, the lines AX, BY, and CZ are concurrent, at a point N called the Jacobi point.

The Jacobi point is a generalization of the Fermat point, which is obtained by letting α = β = γ = 60° and triangle ABC having no angle being greater or equal to 120°.

If the three angles above are equal, then N lies on the rectangular hyperbola given in areal coordinates by

[math]\displaystyle{ yz(\cot B - \cot C) + zx(\cot C - \cot A) + xy(\cot A - \cot B) = 0, }[/math]

which is Kiepert's hyperbola. Each choice of three equal angles determines a triangle center.